There exist \(n\) -dimensional quantum states \(\rho \)  with all diagonal entries \(\frac{1}{n}\)  which do not allow for pure state decomposition \(\lbrace p_k,\ |\psi _k\rangle \rbrace \)  such that all diagonal entries of \(|\psi _k\rangle \langle \psi _k|\)  are \(\frac{1}{n}\)  for all \(k\)  and \(n\ge 4\) . Some specific examples are shown in Refs. [1]}, [2]}. We now give an explicit pure state decomposition for three dimensional AMC states. In a three dimensional system, the quantum state \(\rho =\sum _{i,j} \rho _{ij} |i\rangle \langle j|\) , with \(\rho _{11}=\rho _{22}=\rho _{33}=\frac{1}{3}\)  and real nonzero off diagonal entries, is an example of mixed AMC state that is not a maximally mixed state. Let \(p_1=\frac{1}{4}(1+\rho _{12} +\rho _{13} +\rho _{23} )\) , \(p_2=\frac{1}{4}(1-\rho _{12} -\rho _{13} +\rho _{23} )\) , \(p_3=\frac{1}{4}(1-\rho _{12} +\rho _{13} -\rho _{23} )\) , \(p_4=\frac{1}{4}(1+\rho _{12} -\rho _{13} -\rho _{23} )\) , and
\(|\psi _1\rangle =\frac{1}{\sqrt{3}}(|1\rangle +|2\rangle +|3\rangle )\) , \(|\psi _2\rangle =\frac{1}{\sqrt{3}}(-|1\rangle +|2\rangle +|3\rangle )\) ,
\(|\psi _3\rangle =\frac{1}{\sqrt{3}}(|1\rangle -|2\rangle +|3\rangle )\) ,
\(|\psi _4\rangle =\frac{1}{\sqrt{3}}(|1\rangle +|2\rangle -|3\rangle )\) ,
then \(\lbrace p_k,\ |\psi _k\rangle \rbrace \)  is a pure state decomposition of \(\rho \)  with components all maximally coherent.
